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Saturday, 9 August 2014

chapter 1 concept of limits

The concept of limits forms the basis of calculus and is a very powerful one. Both differential and integral calculus are based on this concept and as such, limits need to be studied in good detail.
This section contains a general, intuitive introduction to limits.
Consider a circle of radius r.
image of a circle
We know that the area of this circle is \pi {r^2} . How?
The ancient Greeks derived this result using the concept of limits.


To see how, recall the definition of \pi .
\begin{array}{l}  \pi  = \dfrac{{{\rm{length}}\,{\rm{of}}\,{\rm{circumference}}}}\,{{{\rm{length}}\,{\rm{of}}\,{\rm{diameter}}}}\,\\  \pi  = \dfrac{c}{d} = \dfrac{c}{{2r}}\\  c = 2\pi r  \end{array}
With this definition in hand, the Greeks divided the circle as follows (like cutting a cake or a pie):
division of a circle by the greeks
Now they took the different pieces of this ‘pie’ and placed them as follows:
image to illustrate the concept of limits
See what happens if the number of cuts are increased
concept of limits
finding the area of a circle
The figure on the right side starts resembling a rectangle as we increase the number of cuts to the circle. The sequence of curves that joins xto y starts becoming more and more of a straight line with the same total length \pi r .


What happens as we increase the number of cuts indefinitely, or equivalently, we decrease \theta  indefinitely? The figure ‘almost’ becomes a rectangle, though never becoming a rectangle exactly. The area ‘almost’ becomes \pi r \times r = \pi {r^2}.
In the language of limits, we say that the figure tends to a rectangle or the area A tends to \pi {r^2} or the limiting value of area is \pi {r^2}.
In standard terminology.
\mathop {{\rm{lim}}}\limits_{\theta  \to 0} {\rm{A}} = \pi {r^2}
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